Online Interval Skyline Queries on Time Series

1. Online Interval Skyline Queries on Time Series Bin Jiang, Jian Pei

2. Outline • Problem Definition • An On-the-fly Method • Interval Skyline Query Answering Algorithm • Online Interval Skyline Query Algorithm • Radix Priority Search Tree • A View-Materialization Method • Non-redundant skyline time series---NRSky[i:j] • Experiments

3. Problem Definition • Notions • Time Series: A time series s consists of a set of ( value, timestamp) pairs.Here we denote the value of s at timestamp I by s[i], and s as a sequence of values s[1],s[2],… • Time Interval: a range in time, denoted as [i : j]. We write if ; if . Some Notions in This Paper

4. Problem Definition • Interval Skyline • Given a set S of time series and interval[i:j], the interval skyline is the set of time series that are not dominated by any other time series in [i:j], denoted by Suppose S={S1, S2, S3} S1 and S2 are in Sky[16:22], while S3 is doninated by S2. S2 S1 S3

5. Problem Definition • Interval Skyline Property 1:If there exist timestamps k1,…,kl(i≤k1<…<kl≤j) such that and s is the only such a time series, then time series is in .

6. Problem Definition • Problem Definition • Given a set of time series S such that each time series is in the base interval ,we want to maintain a data structure D such that any interval skyline queries in interval can be answered efficiently using D. • Methods • An On-The-Fly Method • Original Interval Skyline Query Algorithm • Online Interval Skyline Query Algorithm • A View-Materialization Method

7. Outline • Problem Definition • An On-the-fly Method • Interval Skyline Query Answering Algorithm • Online Interval Skyline Query Algorithm • Radix Priority Search Tree • A View-Materialization Method • Non-redundant skyline time series---NRSky[i:j] • Experiments

8. An Interval Skyline Query Algorithm • Idea Using the maximum value and minimum value of the time series, we can determine the domination of some time series without checking the details.

9. An Interval Skyline Query Algorithm • Algorithm • Set current Skyline Set Sky is null; • Sort the time series in a list L in the descending order of their maximum value; • Set the maximum value of the minimum value of the time series in Sky • For each time series s that satisfies in L, determine whether it can dominate or be dominated by time series in Sky; If it can not be dominated: • add it into Sky ; • delete its dominance in Sky ; • update ; • Return Sky;

10. An Interval Skyline Query Algorithm • Example Goal: compute the skyline in interval [2:3] Steps: 1. s2->Sky, maxmin =1 2. s3->Sky, maxmin =2 3. s5->Sky, maxmin =4 4. s5->s1, s1 is discarded, maxmin =4 5. s4.min=3<4=maxmin, s4 is discarded. Return Sky={s2,s3,s5}

11. An Interval Skyline Query Algorithm • Disadvantage Checking the max value for each time series and the min[i:j] for the query interval [i:j] is costly. • Improvement Idea • Utilize Radix Priority Search Tree to maintain the min[i:j] • Use a sketch to keep the max value for each time series

12. Online Interval Skyline Query Algorithm • Radix Priority Search Tree Radix Priority Search Tree is a two-dimensional data structure, a hybrid of a heap on one dimension and a binary search tree on the other dimension. • Advantages: • Insertion in O(h) • Deletion in O(h) • Query in O(h) • h: the height of the tree

13. Online Interval Skyline Query Algorithm • Radix Priority Search Tree • Build • Use the timestamps as the binary tree dimension X and the data value as the heap dimension Y; • Map W into a fixed domain of X, {0,1,...,w-1}; • The height of the tree is O(logw) • Update → One insertion s[ ] One deletion s[ ] : the most recent timestamp

14. Maintain max values Using Sketches • Sketches • A pair (v,t) is maintained if no other pair (v1,t1) such that v1>v, t1>t; • These pairs form the skyline of points in the interval; • The expected number of points in the skyline is O(logw); • With the sketches, finding the maximum value in W costs O(1) time ; W=[1,3] Sketches : (4,1),(3,2),(2,3) W=[1,4] Sketches : (5,4)

15. Online Interval Skyline Query Algorithm • Complexity • Space • Radix priority search tree O(w) • Sketch of the max values O(logw) Total: O(nw) • Time • Radix priority search tree O(logw) • Sketch of the max values O(logw) Total: O(nlogw)

16. Outline • Problem Definition • An On-the-fly Method • Interval Skyline Query Answering Algorithm • Online Interval Skyline Query Algorithm • Radix Priority Search Tree • A View-Materialization Method • Non-redundant skyline time series---NRSky[i:j] • Experiments

17. A View-Materialization Method • Non-redundant interval skylines A time series s is called a non-redundant skyline time series in interval [i:j] if • S is in the skyline in interval[i:j] • S is not in the skyline in any subinterval[i׳:j׳] [i:j] It can be proved by pigeonhole principle, if there are more than w skyline intervals, at least two of them will share the same starting timestamps, then one of them is not a minimum skyline interval.

18. Useful Theories

19. A View-Materialization Method • Idea Suppose all non-redundant interval skylines are materialized, we can union all these skylines over all intervals in [i:j] and remove those fail Lemma 2. • Algorithm

20. A View-Materialization Method • Example W= [2:4] Goal: compute the interval skyline in [3:4] Steps: 1. s3->Sky 2. s4->Sky 3. s1->Sky(s2 is dominated by s1) Return Sky={s1,s3,s4} How to maintain the non-redundant skylines ?

21. Maintain Non-Redundant Interval Skylines • Steps

22. Maintain Non-Redundant Interval Skylines • Step1 • Use the on-the-fly algorithm to obtain the interval skyline in the new interval W׳. • Find possible false negatives .

23. Maintain Non-Redundant Interval Skylines • Step2-Shared Divide-and-Conquer Algorithm • This algorithm is an extension of the divide-and conquer algorithm(DC). • In SDC, a space is defined as a time interval. Each timestamp represents a dimension. • The related spaces(intervals) are organized as a path, eg. [j:j],[j-1,j],...,[i,j](i<j).

24. Divide-and-Conquer Algorithm Merge Step Divide Step S12 S22 B B S1 S2 B P4 P4 P3 P3 P3 P1 P1 P1 mB P5 P5 P5 P2 P2 P2 S11 S21 mA mA A A A

25. SDC Algorithm • Comparisons • Results

26. Maintain Non-Redundant Interval Skylines • Step3-Remove “redundant time series”

27. Outline • Problem Definition • An On-the-fly Method • Interval Skyline Query Answering Algorithm • Online Interval Skyline Query Algorithm • Radix Priority Search Tree • A View-Materialization Method • Non-redundant skyline time series---NRSky[i:j] • Experiments

28. Experiments • Parameters

29. Experiments • Synthetic Data Sets • Data Sets Properties • Query Efficiency

30. Experiments • Synthetic Data Sets • Update Efficiency • Space Cost

31. Experiments • Stock Data Sets • Query Time

32. Q&A